Abstract: Thirteen integrands that are rational except for the square root of a quartic polynomial with two pairs of conjugate complex zeros are integrated in terms of R-functions of real variables. In contrast with previous tables, the formulas hold for all real intervals of integration for which the integrals exist (possibly as Cauchy principal values). This is achieved by using Landen's transformation and the duplication theorem. In an appendix, an elliptic integral of the third kind with a restricted complex parameter is transformed to make the parameter real. Also, a degenerate integral of the first kind is separated into real and imaginary parts.